Question

topic: inequalities application problem

1. carlos wants to go to the movies and buy snacks. movie tickets cost $12 each, and he wants at least $5 of snacks per movie. he has $100 to spend.

1.write an inequality representing the maximum number of movies carlos can attend while buying at least $5 of snacks per movie.
2.if he attends 6 movies, how much money will he have left?
3.if he decides to spend at least $8 on snacks per movie, how does this change the maximum number of movies he can attend?
ii) a club charges a $50 registration fee and $20 per month. john can spend at most $290 for the first 6 months.

1. write an inequality to determine the maximum number of months john can afford the membership.
2. if john wants to spend at least $240, what is the minimum number of months he must keep the membership?
3. how would the inequality change if the gym offers a 10% discount on the registration fee?

iii) a students phone plan allows 10 gb of data per month. he uses at least 0.5 gb per day.

1. write an inequality for the number of days he can use the phone without exceeding the data limit.

Explanation:

Step1: Define variable for Carlos' problem

Let $m$ be the number of movies Carlos attends. The cost of a movie - ticket is $12$ dollars and the cost of snacks per movie is at least $5$ dollars. His total budget is $100$ dollars. So the inequality is $(12 + 5)m\leq100$, or $17m\leq100$.

Step2: Calculate remaining money for Carlos

If $m = 6$, the total amount he spends is $(12 + 5)\times6=17\times6 = 102$ dollars, which is over - budget. But if we calculate based on the cost structure, the total cost is $17\times6=102$ dollars. Since he has $100$ dollars, he is short by $102-100 = 2$ dollars, or we can say he has $100-102=- 2$ dollars (meaning he doesn't have enough money).

Step3: Adjust for new snack cost for Carlos

If the snack cost per movie is at least $8$ dollars, let $m$ be the number of movies. The inequality is $(12 + 8)m\leq100$, or $20m\leq100$. Solving for $m$, we get $m\leq5$.

Step4: Define variable for John's problem

Let $n$ be the number of months John can afford the membership. The registration fee is $50$ dollars and the monthly fee is $20$ dollars, and his total budget is $290$ dollars. The inequality is $50+20n\leq290$.

Step5: Calculate minimum months for John

If John wants to spend at least $240$ dollars, we set up the equation $50 + 20n\geq240$. First, subtract $50$ from both sides: $20n\geq240 - 50=190$. Then $n\geq\frac{190}{20}=9.5$. Since $n$ represents the number of months, the minimum number of months is $10$.

Step6: Adjust inequality for John with discount

If there is a $10\%$ discount on the registration fee, the new registration fee is $50\times(1 - 0.1)=45$ dollars. The inequality becomes $45+20n\leq290$.

Step7: Define variable for student's problem

Let $d$ be the number of days the student can use the phone. The student uses at least $0.5$ GB per day and the monthly data limit is $10$ GB. The inequality is $0.5d\leq10$.

Answer:

1. For Carlos: $(12 + 5)m\leq100$ (or $17m\leq100$)
2. For Carlos: He doesn't have enough money. He is short by $2$ dollars.
3. For Carlos: $m\leq5$
4. For John: $50+20n\leq290$
5. For John: $10$
6. For John: $45+20n\leq290$
7. For student: $0.5d\leq10$